Planes
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 11.6 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Be able to find the equation of a plane that satisfies certain conditions by finding a
point on the plane and a vector normal to the plane.
• Know how to find the parametric equations of the line of intersection of two (nonparallel) planes.
• Be able to find the (acute) angle of intersection between two planes.
PRACTICE PROBLEMS:
1. For each of the following, find an equation of the plane indicated in the figure.
(a)
(b)
(a) 6x + 4y + 3z = 12; (b) 3x + 2y = 6
For problems 2-6, determine whether the following are parallel, perpendicular,
or neither.
2. Plane P1 : 5x − 3y + 4z = −1 and plane P2 : 2x − 2y − 4z = 9
The planes are perpendicular.
1
3. Plane P1 : 3x − 2y + z = −3 and plane P2 : 5x + y − 6z = 8
The planes are neither parallel nor perpendicular; Detailed Solution: Here
4. Plane P1 : 3x − 2y + z = −3 and plane P2 : −6x + 4y − 2z = 1
The planes are parallel.
→
−
5. Plane P : 5x − 3y + 4z = −1 and line ` (t) = h2 + 2t, 3 − 2t, 5 − 4ti
The plane and the line are parallel.
→
−
6. Plane P : 5x − 3y + 4z = −1 and line ` (t) =
5
3
2 + t, 3 − t, 5 + 2t
2
2
The plane and the line are perpendicular.
7. Give an example of a plane, P , and a line, L, which are neither parallel nor perpendicular to each other.
→
−
→
−
−
−
Suppose your line has the form ` (t) = `0 +t→
v and that your plane has →
n as a normal
−
−
−
−
vector. Then all possible answers are those for which →
v 6k →
n (i.e., →
v 6= c→
n for any
→
−
→
−
→
−
→
−
scalar c) and v 6⊥ n (i.e., v · n 6= 0). The first condition ensures that L and P are
not perpendicular; the second condition ensures that L and P are not parallel.
For problems 8-13, find an equation of the plane which satisfies the given conditions.
8. The plane which passes through the point P (1, 2, 3) and which has a normal vector of
n = 4i − 2j + 6k.
4(x − 1) − 2(y − 2) + 6(z − 3) = 0
→
−
9. The plane which passes through P (−2, 0, 1) and is perpendicular to the line ` (t) =
h1, 2, 3i + th3, −2, 2i.
3(x + 2) − 2y + 2(z − 1) = 0
10. The plane which passes through points A(1, 2, 3), B(2, −1, 5) and C(−1, 3, 3).
−2(x − 1) − 4(y − 2) − 5(z − 3) = 0
11. The plane which passes through A(1, 2, 3) and is parallel to the plane 3x − 5y + z = 2.
3(x − 1) − 5(y − 2) + 1(z − 3) = 0
12. The plane which passes through A(−2, 1, 5) and is perpendicular to the line of intersection of P1 : 3x + 2y − z = 5 and P2 : −y + z = 7.
1(x + 2) − 3(y − 1) − 3(z − 5) = 0; Detailed Solution: Here
2
13. The plane which contains the point A(−2, −1, 3) and which contains the line L : x =
1 + t, y = 3 − 2t, z = 4t.
2(x + 2) − 3(y + 1) − 2(z − 3) = 0
14. Consider the planes P1 : x + y + z = 7 and P2 : 2x + 4z = 6.
(a) Compute an equation of the line of intersection of P1 and P2 .
One parametric equation of the line of intersection is L : x = 3−2t, y = 4+t, z = t
(b) Compute an equation of the plane which passes through the point A(1, 2, 3) and
contains the line of intersection of P1 and P2 .
5(x − 1) + 4(y − 2) + 6(z − 3) = 0
15. Find the acute angle of intersection of P1 : 3x − 2y + 5z = 0 and P2 : −x − y + 2z = 3.
9
−1
√ √
cos
38 6
16. Find the acute angle of intersection of P1 : 3x − 2y − 5z = 0 and P2 : −x − y + 2z = 3.
−11
−1
√ √ ; Detailed Solution: Here
π − cos
38 6
17. Consider the plane which passes through the point Q and whose normal vectors are
parallel to n. And, let P be another point in space, as illustrated below.
(a) Show that the distance between the point P and the given plane is d =
QP · n
|QP · n|
|QP · n|
=
d = kProjn QPk = n
knk =
2
2
knk
knk
knk
3
|QP · n|
.
knk
(b) Use this method to compute the distance between the point P (2, −1, 4) and the
plane x + 2y + 3z = 5.
7
d= √
14
5
18. Consider planes P1 : 2x − 4y + 5z = −2 and P2 : x − 2y + z = 5.
2
(a) Verify that P1 and P2 are parallel.
n1 = h2,
is normal to plane P1 .
−4, 5i 5
is normal to plane P2 .
n2 = 1, −2,
2
Since n1 = 2n2 , we have that n1 and n2 are parallel. And, because these normal
vectors are parallel, the planes P1 and P2 are parallel, too.
(b) Compute the distance between P1 and P2 . (Hint: See the previous problem.)
12
d = √ ; Detailed Solution: Here
45
4